Definition 18 (Hausdorff Measure [1]}) 
Let \(\mathcal {X}\)  be a metric space. For any \(k \in \mathbb {R}_+\) , we define the \(k\) -dimensional Hausdorff measure of a set \(A \subset \mathcal {X}\)  to be
\(2^{-k} \omega _k \lim _{\varepsilon \downarrow 0} \operatorname{vol}_k^\varepsilon (A),\) 
